Symmetrised fractional variation with L1 fidelity for signal denoising via Grünwald-Letnikov scheme

We define, study and implement the model SFV-L^1 : a variational approach to signal analysis exploiting the Riemann-Liouville (RL) fractional calculus. This model incorporates an L^1 fidelity term alongside fractional derivatives of the right and left RL operators to act as regularizers. This approach aims to achieve an orientation-independent protocol. The model is studied in the continuous setting and discretized in 1d by means of a second-order consistent scheme based on approximating the RL fractional derivatives by a truncated Grünwald-Letnikov (GL) scheme. The discrete optimization problem is solved using an iterative approach based on the alternating direction method of multipliers, with guaranteed convergence. A multi-parameter whiteness criterion is introduced which provides automatic and simultaneous selection of the two free parameters in the model, namely the fractional order of differentiation and the regularization parameter. Numerical experiments on one-dimensional signals are presented which show how the proposed model holds the potential to achieve good quality results for denoising signals corrupted by additive Laplace noise.